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(19)
i
i u=1 The geometric interpretation of (17) is the cosine of
the angle p
between thepm-dimensional, unit vectors
;
;p
p^
p1 : : : pm > and q1 : : : qm > .
^
^
^
Using now (17) the distance between two distributions can be de ned asp
^^
d(y) = 1 ; p(y) q] :
(18)
The statistical measure (18) is well suited for the
task of target localization since:
1. It is nearly optimal, due to its link to the Bayes
error. Note that the widely used histogram intersection technique 26] has no such theoretical foundation.
2. It imposes a metric structure (see Appendix). The
Bhattacharyya distance 15, p.99] or Kullback divergence 8, p.18] are not metrics since they violate
at least one of the distance axioms.
3. Using discrete densities, (18) is invariant to the
scale of the target (up to quantization e ects). Histogram intersection is scale variant 26].
4. Being valid for arbitrary distributions, the distance (18) is superior to the Fisher linear discriminant, which yields useful results only for distributions that are separated by the mean-di erence
15, p.132].
Similar measures were already used in computer vision. The Cherno and Bhattacharyya bounds have
been employed in 20] to determine the e ectiveness of
edge detectors. The Kullback divergence has been used
in 27] for nding the pose of an object in an image.
The next section shows how to minimize (18) as a
function of y in the neighborhood of a given location,
by exploiting the mean shift iterations. Only the distribution of the object colors will be considered, although
the texture distribution can be integrated into the same
framework. i=1 where is the Kronecker delta function. The normalization constant C is derived by imposing the condition
Pm
^
u=1 qu = 1, from where
C = Pn k1kx? k2)
(20)
i=1 ( i
since the summation of delta functions for u = 1 : : : m
is equal to one.
Target Candidates Let fxi gi=1:::nh be the pixel
locations of the target candidate, centered at y in the
current frame. Using the same kernel pro le k, but with
radius h,...
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